8 problems

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every sym…

Let $B\geq 0$ be the strength of a constant magnetic field. Let $\Omega^*(B)$ be a minimizing domain, and let $h(\Omega^*(B))$ and $R(\Omega^*(B))$ denote its height and radius. Le…

Let $B\geq 0$ be the strength of a constant magnetic field, and let $\Omega^*(B)$ be a domain minimizing the first eigenvalue among the admissible domains of fixed volume. The magn…

Oja's linear-regret conjecture. There exists such a sequence $\boldsymbol{A}_1,\dots,\boldsymbol{A}_T$, chosen obliviously and independently of Oja's initial random vector, for whi…

Quantitative stability conjecture. There exists $\eta>0$ such that, for every $V\in\mathcal M(\Omega)$,

Ground-state optimization conjecture. The inequality

Let $b_1$ and $b_2$ be constants satisfying $0\le b_1<b_2<+\infty$. For each admissible frequency parameter $\alpha$, let $\beta(\alpha)$ denote the minimal decay rate among the co…

Minimal-energy equality conjecture. The three inequalities in Proposition 1 are equalities.